with the left morphism an isomorphism, the a lift is given by using the inverse of this isomorphism f∘i−1↗{}^{{f \circ i^{-1}}}\nearrow. Hence in particular there is a lift when p∈Kp \in K and so i∈KProji \in K Proj. The other case is formally dual.

and now the top commuting square has a lift by assumption. This is now equivalently a lift in the total diagram, showing that p1∘p1p_1\circ p_1 has the right lifting property against KK and is hence in KInjK Inj. The case of composing two morphisms in KProjK Proj is formally dual. From this the closure of KProjK Proj under transfinite composition follows since the latter is given by colimits of sequential composition and successive lifts against the underlying sequence as above constitutes a cocone, whence the extension of the lift to the colimit follows by its universal property.

By composition, this is also a lift in the total outer rectangle, hence in the original square. Hence jj has the left lifting property against all p∈Kp \in K and hence is in KProjK Proj. The other case is formally dual.

The commutativity of the diagrams that we have established so far shows that the first and second morphisms here equal each other, respectively. By the fact that the universal morphism into a pullback diagram is unique this implies the required identity of morphisms.

Let {(As→isBs)∈KProj}s∈S\{(A_s \overset{i_s}{\to} B_s) \in K Proj\}_{s \in S} be a set of elements of KProjK Proj. Since colimits in the presheaf category𝒞Δ[1]\mathcal{C}^{\Delta[1]} are computed componentwise, their coproduct in this arrow category is the universal morphism out of the coproduct of objects ∐s∈SAs\underset{s \in S}{\coprod} A_s induced via its universal property by the set of morphisms isi_s: